New York State Common Core
Mathematics Curriculum
PRECALCULUS AND ADVANCED TOPICS • MODULE 5
Table of Contents1
Probability and Statistics
Module Overview .................................................................................................................................................. 2
Topic A: Probability (S-CP.B.8, S-CP.B.9) ............................................................................................................... 7
Lesson 1: The General Multiplication Rule ............................................................................................... 8
Lesson 2: Counting Rules—The Fundamental Counting Principle and Permutations ........................... 23
Lesson 3: Counting Rules―Combinations .............................................................................................. 37
Lesson 4: Using Permutations and Combinations to Compute Probabilities………………………………………49
Topic B: Random Variables and Discrete Probability Distributions (S-MD.A.1, S-MD.A.2, S-MD.A.3,
S-MD.A.4) .............................................................................................................................................. 64
Lesson 5: Discrete Random Variables..................................................................................................... 66
Lesson 6: Probability Distribution of a Discrete Random Variable......................................................... 80
Lesson 7: Expected Value of a Discrete Random Variable ..................................................................... 92
Lesson 8: Interpreting Expected Value ................................................................................................. 103
Lessons 9–10: Determining Discrete Probability Distributions ............................................................ 113
Lessons 11–12: Estimating Probability Distributions Empirically ......................................................... 132
Mid-Module Assessment and Rubric ................................................................................................................ 155
Topics A through B (assessment 1 day, return 1 day, remediation or further applications 1 day)
Topic C: Using Probability to Make Decisions (S-MD.B.5, S-MD.B.6, S-MD.B.7) .............................................. 168
Lessons 13–14: Games of Chance and Expected Value........................................................................ 170
Lesson 15: Using Expected Values to Compare Strategies ................................................................... 190
Lesson 16: Making Fair Decisions ......................................................................................................... 203
Lesson 17: Fair Games .......................................................................................................................... 210
Lessons 18–19: Analyzing Decisions and Strategies Using Probability ................................................. 221
End-of-Module Assessment and Rubric ............................................................................................................ 244
Topics A through C with a focus on Topic C (assessment 1 day, return 1 day, remediation or further
applications 1 day)
1
Each lesson is ONE day, and ONE day is considered a 45-minute period.
Module 5:
Date:
Probability and Statistics
2/8/16
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview
M5
PRECALCULUS AND ADVANCED TOPICS
Precalculus • Module 5
Probability and Statistics
OVERVIEW
In this module, students build on their understanding of probability developed in previous grades. In Topic A,
the multiplication rule for independent events introduced in Grade 11 is generalized to a rule that can be
used to calculate the probability of the intersection of two events in situations where the two events are not
independent. In this topic, students are also introduced to three techniques for counting outcomes—the
fundamental counting principle, permutations, and combinations. These techniques are then used to
calculate probabilities, and these probabilities are interpreted in context (S-CP.B.8, S-CP.B.9).
In Topic B, students study probability distributions for discrete random variables (S-MD.A.1). They develop an
understanding of the information that a probability distribution provides and interpret probabilities from the
probability distribution of a discrete random variable in context (S-MD.A.2). For situations where the
probabilities associated with a discrete random variable can be calculated given a description of the random
variable, students determine the probability distribution (S-MD.A.3). Students also see how empirical data
can be used to approximate the probability distribution of a discrete random variable (S-MD.A.4). This topic
also introduces the idea of expected value, and students calculate and interpret the expected value of
discrete random variables in context.
Topic C is a capstone topic for this module, where students use what they have learned about probability and
expected value to analyze strategies and make decisions in a variety of contexts (S-MD.B.5, S-MD.B.6,
S-MD.B.7). Students use probabilities to make a fair decision and explain how to make fair and “unfair”
decisions. Students analyze simple games of chance as they calculate and interpret the expected payoff in
context. They make decisions based on expected values in problems with business, medical, and other
contexts. They also examine and interpret what it means for a game to be fair. Interpretation and
explanations of expected values are important outcomes for Topic C.
Focus Standards
Use the rules of probability to compute probabilities of compound events in a uniform
probability model.
S-CP.B.8
(+) Apply the general Multiplication Rule in a uniform probability model, 𝑃(𝐴 and 𝐵) =
𝑃(𝐴)𝑃(𝐵|𝐴) = 𝑃(𝐵)𝑃(𝐴|𝐵), and interpret the answer in terms of the model.
S-CP.B.9
(+) Use permutations and combinations to compute probabilities of compound events and
solve problems.
Module 5:
Date:
Probability and Statistics
2/8/16
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Module Overview
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
Calculate expected values and use them to solve problems.
S-MD.A.1
(+) Define a random variable for a quantity of interest by assigning a numerical value to each
event in a sample space; graph the corresponding probability distribution using the same
graphical displays as for data distributions.
S-MD.A.2
(+) Calculate the expected value of a random variable; interpret it as the mean of the
probability distribution.
S-MD.A.3
(+) Develop a probability distribution for a random variable defined for a sample space in
which theoretical probabilities can be calculated; find the expected value. For example, find
the theoretical probability distribution for the number of correct answers obtained by
guessing on all five questions of a multiple-choice test where each question has four choices,
and find the expected grade under various grading schemes.
S-MD.A.4
(+) Develop a probability distribution for a random variable defined for a sample space in
which probabilities are assigned empirically; find the expected value. For example, find a
current data distribution on the number of TV sets per household in the United States, and
calculate the expected number of sets per household. How many TV sets would you expect
to find in 100 randomly selected households?
Use probability to evaluate outcomes of decisions.
S-MD.B.5
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and
finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected
winnings from a state lottery ticket or a game at a fast-food restaurant.
b.
Evaluate and compare strategies on the basis of expected values. For example,
compare a high-deductible versus a low-deductible automobile insurance policy using
various, but reasonable, chances of having a minor or a major accident.
S-MD.B.6
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
S-MD.B.7
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a game).
Foundational Standards
Understand independence and conditional probability and use them to interpret data.
S-CP.A.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of other events
(“or,” “and,” “not”).
S-CP.A.2
Understand that two events 𝐴 and 𝐵 are independent if the probability of 𝐴 and 𝐵 occurring
together is the product of their probabilities, and use this characterization to determine if
they are independent.
Module 5:
Date:
Probability and Statistics
2/8/16
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Module Overview
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
S-CP.A.3
Understand the conditional probability of 𝐴 given 𝐵 as 𝑃(𝐴 and 𝐵)/𝑃(𝐵), and interpret
independence of 𝐴 and 𝐵 as saying that the conditional probability of 𝐴 given 𝐵 is the same
as the probability of 𝐴, and the conditional probability of 𝐵 given 𝐴 is the same as the
probability of 𝐵.
S-CP.A.4
Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to
decide if events are independent and to approximate conditional probabilities. For example,
collect data from a random sample of students in your school on their favorite subject
among math, science, and English. Estimate the probability that a randomly selected
student from your school will favor science given that the student is in tenth grade. Do the
same for other subjects and compare the results.
S-CP.A.5
Recognize and explain the concepts of conditional probability and independence in everyday
language and everyday situations. For example, compare the chance of having lung cancer if
you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform
probability model.
S-CP.B.6
Find the conditional probability of 𝐴 given 𝐵 as the fraction of 𝐵’s outcomes that also
belong to 𝐴, and interpret the answer in terms of the model.
S-CP.B.7
Apply the Addition Rule, 𝑃(𝐴 or 𝐵)– 𝑃(𝐴) + 𝑃(𝐵)– 𝑃(𝐴 and 𝐵), and interpret the answer in
terms of the model.
Focus Standards for Mathematical Practice
MP.2
Reason abstractly and quantitatively. Students interpret probabilities calculated using the
addition rule and the general multiplication rule. They use permutations and combinations
to calculate probabilities and interpret them in context. Students also explain the meaning
of the expected value of a random variable as a long-run average and connect this
interpretation to the given context.
MP.3
Construct viable arguments and critique the reasoning of others. Students construct
arguments in distinguishing between situations involving combinations and those involving
permutations. Students use permutations and combinations to calculate probabilities and
evaluate decisions based on probabilities. Students also use expected values to analyze
games of chance and to evaluate whether a game is “fair.” Students design, compare, and
evaluate games of chance that they construct, comparing their games to the games of other
students based on probabilities and expected values. They analyze strategies based on
probability. For example, students use expected value to explain which of two plans yields
the largest earnings for an insurance company.
MP.4
Model with mathematics. Students develop a probability distribution for a random variable
by finding the theoretical probabilities. Students model probability distributions by
estimating probabilities empirically. They use probabilities to make and justify decisions.
Throughout the module, students use statistical ideas to explain and solve real-world
Module 5:
Date:
Probability and Statistics
2/8/16
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview
M5
PRECALCULUS AND ADVANCED TOPICS
problems. For example, given the probability of finding a female egg in a nest, students
determine a discrete probability distribution for the number of male eggs in the nest.
MP.5
Use appropriate tools strategically. Students use technology to carry out simulations in
order to estimate probabilities empirically. For example, students use technology to
simulate a dice-tossing game and generate random numbers to simulate the flavors in a
pack of cough drops. They use technology to graph a probability distribution and to
calculate expected values. Students come to view discrete probability distributions as tools
that can be used to understand real-world situations and solve problems.
MP.8
Look for and express regularity in repeated reasoning. Students use simulations to observe
the long-run behavior of a random variable, using the results of the simulations to estimate
probabilities.
Terminology
New or Recently Introduced Terms
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Combination of 𝒌 items selected from a set of 𝒏 distinct items (an unordered set of 𝑘 items
selected from a set of 𝑛 distinct items).
Continuous random variables (a random variable for which the possible values form an entire
interval along the number line).
Discrete random variables (a random value for which the possible values are isolated points along
the number line).
Empirical probability (a probability that has been estimated by observing a large number of
outcomes of a chancge experiment or values of a random variable).
Expected value of a random variable (the long-run average value expected over a large number of
observations of the value of a random variable).
Fundamental counting principle (Let 𝑛𝟏 be the number of ways the first step or event can occur and
𝑛𝟐 be the number of ways the second step or event can occur. Continuing in this way, let 𝑛𝒌 be the
number of ways the 𝑘 𝑡ℎ stage or event can occur. Then, based on the fundamental counting
principle, the total number of different ways the process can occur is 𝑛𝟏∗𝑛𝟐∗𝑛𝟑∗….∗𝑛𝒌.)
General multiplication rule (a probability rule for calculating the probability of the intersection of
two events).
Long-run behavior of a random variable (the behavior of the random variable over a very long
sequence of observations).
Permutation of 𝒌 items selected from a set of 𝒏 distinct items (an ordered sequence of 𝑘 items
selected from a set of 𝑛 distinct items).
Probability distribution (a table or graph that provides information about the long-run behavior of a
random variable).
Probability distribution of a discrete random variable (a table or graph that specifies the possible
values of the random variable and the associated probabilities).
Random variable (a variable whose possible values are based on the outcome of a random event).
Module 5:
Date:
Probability and Statistics
2/8/16
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Module Overview
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
PRECALCULUS AND ADVANCED TOPICS
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Theoretical probability (a probability calculated by assigning a probability to all possible outcomes in
the sample space for a chance experiment).
Uniform probability model (a probability distribution that assigns equal probability to each possible
outcome of a chance experiment).
Familiar Terms and Symbols2
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Chance experiment
Complement of an event
Event
Intersection of events
Sample space
Union of events
Suggested Tools and Representations
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Graphing calculator or graphing software
Random number software
Random number tables
Two-way frequency tables
Assessment Summary
Assessment Type
Administered Format
Standards Addressed
Mid-Module
Assessment Task
After Topic B
Constructed response with rubric
S-CP.B.8, S-CP.B.9,
S-MD.A.1, S-MD.A.2,
S-MD.A.3, S-MD.A.4
Constructed response with rubric
S-CP.B.8, S-CP.B.9,
S-MD.A.2, S-MD.A.3,
S-MD.B.5, S-MD.B.6,
S-MD.B.7
End-of-Module
Assessment Task
2
After Topic C
These are terms and symbols students have seen previously.
Module 5:
Date:
Probability and Statistics
2/8/16
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